On the topic of matrix equation solvers, the Harrow/Hassidim/Lloyd (HHL) algorithm [1]} has become well established as the leading quantum algorithm.
For a system of condition number \(\kappa \) , this algorithm is capable of providing computational complexity \(\mathcal {O}\left(\kappa ^2\log \left(N\right)/\epsilon \right)\)  in the calculation of a solution of precision \(\epsilon \) .
Thus, when \(\kappa \)  and \(1/\epsilon \)  scale logarithmically with \(N\) ,
the procedure offers an exponential speedup over classical algorithms.
